I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up- Intercepts of Polynomial Functions, which asks students to find a polynomial given two roots.
I also use this time to correct and record the previous day's Homework.
Identifying Roots From A Graph
We begin by finding the zeros of a polynomial function graphically using a calculator which is a review of the previous lesson. The goal of this is to give a jumping off point for the rest of the lesson.
Now we look at roots that occur twice in a polynomial (have a multiplicity of 2). The students graph x4 – 5x3 – 7x2 + 29x + 30 = 0, which has a root with a multiplicity of two (I DO NOT warn them of this in advance), using a graphing calculator, and identify its zeros. I ask them to write this as a product of binomials using those zeros. The students probably write (x+1)(x-1)(x-2)=0. I point out that this only gives us a cubic and we want a fourth degree polynomial. The students look at the graph again and write a theory as to the location of the missing zero (discussing in partners works well here). I love problems like this that don't work the way students expect them to. Using a bit of flare, this kind of "mystery" can really engage students.
To check their theory or get some more hints, I remind them that they can always test other polynomials. Some will get overwhelmed at the idea of finding other polynomials with this pattern, so I tell them that the factored form of a polynomial works just as well as the extended form when graphing in a calculator. There may be a few that will need me to model this technique for them.
Once most of the students have a theory, we discuss it as a class. By the end, the students should have a statement written in their notes regarding the shape of double zeros in a graph. (This will be useful in the next couple of lessons.) I chose not to deal with zeros that happen more than twice as it goes beyond the purpose of today's lesson.
Writing Polynomials from a Graph
The next task involves a problem that gives the students a graph and asks them to write a polynomial equation using the roots of the graph. It has a zero with a multiplicity of two. The equation they end up with will not actually recreate this particular graph but we don’t address this at this time. Please be aware, I am not dealing with the shrink factor yet. What they get will have the same zeros but will have a much smaller local minimum. This problem will come back up later in the lesson when we will deal with the stretch.
Finally, we discuss imaginary solutions. The students find the zeros of each polynomial using algebra. I ask them to predict what those answers will look like on the graph. Then they graph each polynomial and write a statement about how those imaginary zeros are represented graphically. They can test additional polynomials with imaginary solutions if necessary. The goal is that the students recognize that imaginary solutions do not provide x-intercepts. There may be an extra curve above or below the x-axis but that isn’t always the case. We then discuss if there can only be one imaginary solution.
For detailed presentation notes, please see the PowerPoint.
The first task in this section is to relate how zeros are important to polynomial graphs. They saw this idea in quadratics but so often students struggle with taking ideas in mathematics and extending them to other places. They discuss this as pairs and write a statement in their notes. We then discuss this as a class as well. The key here is linking the term zero to x-intercept. Please check out the great article from Edutopia on classroom discussions.
The next step is for the students to find the y-intercept. This step will be important as students learn to find the equations of polynomials given the graph. I give them some time to find the y-intercept given the extended form. Once they have it, I give them the factored form and ask them to locate the y-intercept of the polynomial without factoring. There is an additional practice problem and I may add additional problems as needed.
We are now going back to the graph from the previous section (Types of Zeros of Polynomial Equations). The students found the zeros but haven't checked those zeros against the graph. The students graph the polynomial that they wrote with these zeros on a calculator. Next, they verbally compare with their partner the graph on the PowerPoint slide to their graph. They should notice that the graphs match x-intercepts but not curves. I tell them: “This must be some kind of transformation, which one would it be?” “Where would it be located in the function? “ They then experiment by putting in a multiple to see if they can reproduce this graph (Math Practice 1). A good scaffolding hint would be to have them look at the y-intercept.
Next, they are given another graph and asked to find the equation (which has a stretch of 3). I encourage them to test the different stretches until they find one that works.
The next problem asks them to find an equation from a graph WITHOUT using a calculator and then describe their process. There are several methods. I am not planning on instructing them to do one particular way. Rather, I will highlight some of the good thinking of their peers and let them pick what works best for them. With that being said, a very intuitive method that will work for many students is to find the y-intercept of the function with no stretch/shrink and then find the number that they can multiply on to that intercept to get the y-intercept of the pictured graph.
The final practice problem gives the students a graph with a ½ shrink. Some will get stuck on how to represent divide 2 as a number. I know they should understand, but a few of my students still struggle with fractions and their relationship to division. Using other students to explain this type of transformation is a good scaffolding method (Math Practice 3).
Please see the PowerPoint for detailed presentation notes.
The first 6 problems ask the students to find the x- and y-intercepts of polynomials. Half are extended polynomials and half are factored. These problems provide a range of difficulty and can be easily differentiated to support a large variety of students. The final four problems present graphs and ask for appropriate equations including the proper stretch or shrink. These really focus on both Math Practice 7 and Math Practice 1 as many won't have a graphing calculator to check their equations.
This assignment was created using Kuta Software, an amazing resource for secondary mathematics teachers.
I use an exit ticket each day to provide a quick formative assessment to judge the success of the lesson.
This Exit Tickets is a snap shot that identifies how many students can find the x- and y-intercepts of a factored cubic.